Ten things, one phenomenon?

Two were cheap enough to pilot straight away. The other two need a bigger machine or a real lab.

The corner. "Twisted" means a field comes back as its negative after one trip around (a Z2 holonomy, the Möbius strip's −1). "Chaotic" means nearby states separate exponentially. The corner holds systems that are both, where the twist shapes the chaos.

What PG-026 left behind. In round 4 the Möbius fluid stayed calm where the plain loop of the same length went chaotic. It behaved like a loop twice as long. In round 5 the full fluid at r = 150 was chaotic but never flipped. So every twisted system built so far either sat still or copied its double.

What counts as a resident. A twisted system whose chaos shows something neither control shows. The controls are the untwisted loop of the same length and the untwisted loop of twice the length. It must survive a finer grid, five seeds and a window of parameters; one lucky point does not count.

Ben's two doors. Ben named two places to look. "Spinning fluids with chaotic spin fields" became a quantum fluid whose spin carries a half twist (E3), with a classical spinning fluid as the fallback. "3D flow on non-orientable domains" became convection in a box glued like a Klein bottle (E4). E1 and E2 are cheap warm-ups that test the method.

The double-cover trap

Every twisted loop of length L is hiding inside a plain loop of length 2L. That one fact killed round 4, so every experiment here is built around it.

A field that flips after one lap is the same thing as a field on a loop twice as long that is forced to be its own negative halfway round:

u(x + L) = −u(x)   ⇔   u lives on a loop of length 2L and keeps only its odd modes

So the twisted system is a slice (an invariant subspace) of its untwisted double. Two things can happen to that slice inside the double:

• The slice attracts. The double falls into the twisted state on its own. The twist only made the box longer. This is what round 4 saw: the Möbius fluid matched the 2L loop.
• The slice repels. Inside the double, a tiny untwisted nudge grows and the state leaves the slice. Now the twisted chaos is something the double never shows you unaided. That is the only door to a resident.

The number that decides it is the transverse exponent λ⊥: the growth rate of an untwisted nudge to the twisted state, measured inside the double. This is the same quantity behind blowout bifurcations and on-off intermittency (Ott and Sommerer, 1994; cited from memory).

Every experiment runs the same three-way control:

The runner and its six gates

A proposal works only if six things are true at once. Each is a gate with a number on it. The first gate that fails names the finding.

All six true: Resident.

Read the table top down: each "yes" takes the run one gate further; each "no" stops it and names the finding.

What comes out of every run, resident or not:

• A resident card (JSON): λ1 for CYL-L, MOB-L and CYL-2L with seed spread, λ⊥, odd and even decay rates, the gate that failed, and the verdict.
• A plate: the three runs side by side as space-time pictures, in the notebook's style, so the difference shows before any number is read.
• A PREREG hash locked before the run, as in rounds 1 to 5.

If a resident turns up, three more things come out: a short methods note (what the twist does that the double cannot), a page of its own on this site, and a protocol a real lab could run (E3 and E4 each name one).

E1 Lorenz, a corner already occupied

The butterfly is already a twisted chaotic system, so it is the first door to knock on. The flip (x, y, z) → (−x, −y, z) swaps the two wings. Fold the wings together and you get a one-lobe system (proto-Lorenz, Miranda and Stone 1993). Lorenz is its double cover (Letellier and Gilmore, 2000). Every wing switch is the −1 picked up going round. Both citations are from memory and were never re-opened.

The question. Does the wing label carry chaos of its own, or does the folded system decide every wing in advance?

Controls. The folded base (proto-Lorenz coordinates), the cover (Lorenz itself), and the cover with the flip broken by a small push ε added to dx/dt.

Needed for a resident

• The wing sequence must hold information the base doesn't. Check: the entropy rate of the left/right sequence must exceed that of the stay/switch sequence. Equal means the base decides every wing (Shadow).
• Odd quantities (x, y, xz, yz) must lose memory at a different rate from even ones (z, x², xy, y²). This is gate T5.
• Sanity: λ1 must be the same in both coordinates. This is a theorem, so failing it means a bug.

What comes out. Three numbers: λ1 in both coordinates, the odd-to-even memory-loss ratio, and the two entropy rates. A plate: the butterfly with each lap inked by wing, beside its folded one-lobe base.

Locked prediction. Shadow. The bet: the twist changes how fast the system forgets and leaves alone how fast nearby paths separate.

E2 A ring of oscillators that must come back upside down

The complex Ginzburg–Landau equation is the cleanest chaotic field with a legal flip, so it calibrates the method. It describes any medium of coupled oscillators near onset: chemical waves, lasers, convection rolls.

∂A/∂t = A + (1 + i c1) ∂²A/∂x² − (1 + i c2) |A|² A,   with   A(x + L) = −A(x)

The twist. Turning every oscillator by half a turn (A → −A) leaves the equation unchanged, so the twisted ring is legal (gate T1). It forces the phase to turn by an odd number of half-turns per lap. A plain ring can sit at zero turn; the twisted one never can.

Why this regime. In phase turbulence the winding number is conserved and shapes the chaos: rings wound past a critical value νc decay by phase slips, and the allowed band shrinks to zero at the defect-turbulence line (Montagne, Hernández-García and San Miguel, PRL 77, 267, 1996; Montagne et al., PRE 56, 151, 1997). The pilot uses their parameters: c1 = 2.1, c2 = −0.75 and −0.83, with the defect line near c2 = −0.89.

Needed for a resident

• The twisted ring must be chaotic (T2), and keep its half-turn winding without phase slips (T1).
• The gap to the generic double (CYL-2L) must hold its size as the ring grows (T3). If it shrinks like 1/L, the twist is only a phase gradient of π/L spread thin.
• The double must throw out the twisted state (T4, λ⊥ > 0).

What comes out. A table of λ1 for the three rings at L = 32, 64, 128, 256 with 5 seeds each, λ⊥, and the distance between amplitude histograms. A plate: three space-time strips of |A| side by side.

Locked prediction. Shadow, with the twist's effect fading like 1/L.

Where a resident could still hide here (E2b, still to run). Near the defect line νc shrinks toward zero. A plain ring can sit at ν = 0 and stay calm-ish. The twisted ring is forced to carry winding, which may push it over into defect turbulence first. That would be chaos the twist itself switched on. The fair control is then the double started at winding 1.

E3 Spinning fluids: a quantum fluid whose spin carries a half twist

This is the door where the double-cover trap may not close. Here the twist lives in the fluid itself, so there is no fixed double to hide in.

The fluid. A spin-1 sodium condensate in its polar phase. Its order parameter is a phase angle θ plus a spin axis d. The pair (θ + π, −d) is the same state as (θ, d). That identification is the Z2 twist, built into the material.

The resident candidate. A half-quantum vortex: the phase turns by half a turn and the spin axis turns by half a turn, together closing up. It is an "Alice string": carry a spin round one and it comes back flipped. Seo, Kwon, Kang and Shin imaged them in exactly this system (PRL 115, 015301, 2015).

Why the trap may not close. To make a half vortex single-valued you lift the field onto a double cover of the plane, cut open at every vortex core. Every time vortices move, pair up or annihilate, that cover changes shape. Round 4's fix ("compare with the 2L box") has nothing fixed to compare with.

The run. A 2D three-component Gross–Pitaevskii simulation, 256 × 256, stirred into a tangle of many half vortices. The "spin field" goes chaotic as they collide.

Controls.

Needed for a resident

• Half vortices must survive long enough to tangle before pairing into whole vortices.
• The tangle must be chaotic: λ1 > 0 by twin runs (T2).
• TWIST must differ from COVER: annihilation rate, energy spectrum or λ1 (T3).
• The difference must sit in odd quantities: the spin axis d itself (odd) against its sign-blind tensor dd (even). This is the same split as E1 (T5).
• It must hold at 512 × 512 and across a range of the magnetic field term q (T6).

What comes out. λ1, half-vortex count over time and pair-annihilation rate for all three runs, plus odd and even spin correlations. A plate: the spin axis as a colour wheel with half vortices as dots, TWIST beside COVER.

If a resident turns up, the room to stand in. A sodium spinor lab like Shin's at Seoul National University: quench to make many half vortices, image the spin components separately, count annihilations against q. The protocol would be written so they could run it as is.

Cost. A small version runs in about an hour (an estimate). The full scan needs a GPU.

Classical fallback. A fluid of spinning particles (an odd-viscosity chiral fluid) on a Möbius strip. Spin is handed, and handedness flips once round the strip. So a uniformly spinning flock cannot exist there: it must grow a seam where the spin passes through zero. That is both doors in one system.

E4 Flow on non-orientable domains

One hard fact first: no 3D box in ordinary space is non-orientable. Every region of ordinary space inherits its left and right hands. A twisted tube looks like a Möbius band, but as a solid it is orientable. So 3D flow on a non-orientable domain exists only in simulation. A 2D flow on a non-orientable surface can be built on a bench. E4 has one arm of each.

E4a Convection in a Klein box (simulation only)

Glue the ends of a convection box with a mirror: (x + L, y, z) is the point (x, −y, z), and the y-velocity flips sign. Gravity keeps pointing down, so heating from below stays legal (T1).

The twist bites the big roll. Large convection cells grow one domain-filling roll that wanders and sometimes reverses (Sugiyama et al., PRL 105, 034503, 2010, cited from memory). A roll turning in the y–z plane cannot survive the mirror gluing: its sense must reverse somewhere along x. So the twist forces a seam into any roll of that orientation, just like the heat seam on Plate III. The prediction: reversals stop being whole-cell flips and become seams sliding along x.

Needed. The plain box must show a big roll that reverses within the run time; without that there is nothing to change. The seam must move rather than pin to one place (T2). Reversal statistics must differ from the 2L double, with the double throwing out the mirrored state (T3, T4).

What comes out. Reversals per 1,000 free-fall times for the three boxes, the seam's position over time, λ1 and λ⊥. A plate: the mid-height temperature drawn on a Klein-bottle surface, with the seam in red.

Cost. Days per run at Rayleigh numbers near 10⁷ (an estimate; a short timing run comes first).

E4b 2D turbulence on a Klein bottle (cheap, and buildable)

In a flat 2D flow, spin (vorticity) is handed. Glue a rectangle's ends with a mirror and a vortex carried once round comes back spinning the other way: ω(x + L, y) = −ω(x, −y).

Why it is the best bet. Forced 2D turbulence feeds its energy upward into one giant flow, a condensate. On some rectangles that condensate flips at random between a vortex pair and straight jets (Bouchet and Simonnet, PRL 102, 094504, 2009, cited from memory). The mirror gluing changes which giant flows are allowed. It permits one with twice the wavelength and forbids others. So the twist should reshape the random switching itself.

Needed. The plain torus must switch at the chosen aspect ratio and run length. The Klein run must still switch (T2). Its switching rate or its set of condensates must differ from the 2L double's (T3). Inside the double, a nudge off the mirrored slice must grow (T4).

What comes out. A switching-time histogram for all three domains, condensate shapes, and λ⊥. A plate: vorticity on the Klein rectangle, with the arrows that show the gluing.

The room to stand in. A soap film stretched on a Möbius wire frame is a real non-orientable 2D fluid. Soap films are already a standard bench for 2D turbulence. Comb it or blow on it, film it, and track whether big vortices come back reversed. This is the one experiment here that fits on a kitchen table.

Cost. A 2D spectral code at 256 × 256 is cheap, though rare switches need long runs: hours to a day.

Pilot results

No resident yet. Both cheap pilots landed where the bets said, one gate earlier than expected. Five of seven locked predictions held (PREREG-6, sha 0bd96df3).

E1 verdict: Shadow of the base. The folded system decides every wing, so the flip adds no chaos. It does change memory: odd quantities forget 12.6× faster. But breaking the flip (ε up to 5) leaves that ratio at 8 to 14. So the fast forgetting comes from wing switching; the exact twist is not needed for it.

E2 verdict: Side room. The twisted ring is up to 10× calmer than both plain rings at L ≤ 128. A control added after the first two sizes (labelled exploratory in the notes) explains it: the 2L ring wound once is just as calm. The twist acts through the winding it forces, and the double can carry that winding too.

Largest Lyapunov exponent, five seeds per point, c1 = 2.1, measured over 1,500 time units:

The twisted ring tracks the wound double from L = 64 up, and both rejoin the plain rings by L = 256. One point breaks the pattern: at L = 32, c2 = −0.83 the twisted ring is calm (λ1 = 0.0006 ± 0.0007) while the wound double is chaotic (0.0148 ± 0.0047). One point is a lead and no more. A fine scan of small rings is the next run.

Lesson for the runner. Gate T3 must compare against the double in the same winding sector; a generic double is too easy to beat. Without that, E2 would have passed T3 at L = 256 on a 20% gap that winding alone explains.

Notes on the run. Before any result, a time-stepper bug was found and fixed (a half-circle contour, valid only for real operators). That first launch was discarded.

Ranking: where to knock next

Knock first on the small twisted rings, because that is the only lead on the board and it costs an hour. Then the Klein-bottle turbulence, which is the best cheap shot at a real resident.

What would change this order. If the E2 rescan shows a window of twisted calm (T6), it jumps to a full card. If E4b's plain torus will not switch at a reachable run length, E3 moves up.